We identify a new secular instability of eccentric stellar disks around supermassive black holes. We show that retrograde precession of the stellar orbits, due to the presence of a stellar cusp, induces coherent torques that amplify deviations of individual orbital eccentricities from the average, and thus drive all eccentricities away from their initial value. We investigate the instability using N-body simulations, and show that it can propel individual orbital eccentricities to significantly higher or lower values on the order of a precession timescale. This physics is relevant for the Galactic center, where massive stars are likely to form in eccentric disks around the SgrA* black hole. We show that the observed bimodal eccentricity distribution of disk stars in the Galactic center is in good agreement with the distribution resulting from the eccentricity instability and demonstrate how the dynamical evolution of such a disk results in several of its stars acquiring high (1 - e a?? 0.1) orbital eccentricity. Binary stars on such highly eccentric orbits would get tidally disrupted by the SgrA* black hole, possibly producing both S-stars near the black hole and high-velocity stars in the Galactic halo.